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TestofHypothesisaroundsinglemean.docx

Test of Hypothesis around single mean

Steps for conducting the test of hypothesis.

1. Set up both null and alternate hypothesis:

H0: =  vs. H1: (always inequality); The options could be or, or,

2. = whatever number given in the problem: rejection region

3. Test Statistics: Could be z or t, based on the sample size. We use z for large sample and t for small sample

z = (x bar ) n or, t = (x bar ) sn.

If we do not know in case of large sample, we replace with s in our formula for z.

4 Rejection Region: The rejection region is dependent on H1. -- something, the whole rejection region goes to the right of the tail. If something, the whole rejection region goes to the left of the tail and something, we divide the rejection region into two equal parts, and each goes both side of the tail. This is similar to confidence interval.

5. Decision Criteria: We compare our computed value from Step 3 with the rejection region value from step 4 and make our decision.

Test of Hypothesis: Sample problems

1. A statewide real estate sales agency specializes in selling farm properties in the state of Ohio. Their records indicate that the mean selling time of a farm property is 90 days. Because of recent drought conditions, they believe that mean selling time now greater than 90 days. . A statewide survey of 100 farms sold recently revealed a mean selling time of 94 days with a standard deviation of 22 days. At the 10% significance level, has there been an increase in selling time?

  µ = 90, s = 22, n = 100, x¯ = 94, = .1

 

H0: =90 vs. H1: >90

= .1

Critical value of z = 1.285 From z-table

z = X¯ - µ = 94 – 90 = 42.2 = 1.82

s/√n 22/√100

H0 is rejected because 1.82 is less than or equal to 1.285. We can conclude that

2. Listed below is the rate of return for one year (reported in percent) for a sample of 12 mutual funds that are classified as taxable money market funds.

4.63 4.15 4.76 4.70 4.65 4.52 4.70 5.06 4.42 4.51 4.24 4.52

Is it reasonable to conclude that the mean rate of return is more than 4.50 percent at a 95% confidence level?

s = .24, X¯ = 4.5717, n = 12, s = .24, α = .05, df = 11

H0: = 4.5 vs. H1: > 4.5

= .05

t = X¯ – µ = 4.5717 – 4.5 = .0717 = 1.03

s/√n . 24 / √12 .0696

Reject H0 if t > 1.796

Do not reject H0 as computed t < 1.796 (From t-table for df = 11 and = .05). We cannot conclude that the mean rate of return is more than 4.5%.

Type I vs. Type II error

It is important to mention that there are two types of possible errors when doing hypothesis testing; the Type I or alpha (α) and the Type II or beta (β). The first error happens when the Ho is rejected even though it is true. The second error happens when we accept the Ho when it is false. The following chart taken from our readings will clarify these two errors:

Null Hypothesis

Accepts Ho

Rejects Ho

Ho is true

Correct decision

Type I error

Ho is false

Type II error

Correct decision

Sample Size

The best way to figure this one out is to think about it backwards. Let's say you picked a specific number of people in the United States at random. What then is the chance that the people you picked do not accurately represent the U.S. population as a whole? For example, what is the chance that the percentage of those people you picked who said their favorite color was blue does not match the percentage of people in the entire U.S. who like blue best?

(Of course, our little mental exercise here assumes you didn't do anything sneaky like phrase your question in a way to make people more or less likely to pick blue as their favorite color. Like, say, telling people "You know, the color blue has been linked to cancer. Now that I've told you that, what is your favorite color?" That's called a leading question, and it's a big no-no in surveying.)

Common sense will tell you (if you listen...) that the chance that your sample is off the mark will decrease as you add more people to your sample. In other words, the more people you ask, the more likely you are to get a representative sample. This is easy so far, right?

Okay, enough with the common sense. It's time for some math. The formula that describes the relationship I just mentioned is basically this:

The margin of error in a sample = 1 divided by the square root of the number of people in the sample

How did someone come up with that formula, you ask? Like most formulas in statistics, this one can trace it roots back to pathetic gamblers who were so desperate to hit the jackpot that they'd even stoop to mathematics for an "edge." If you really want to know the gory details, the formula is derived from the standard deviation of the proportion of times that a researcher gets a sample "right," given a whole bunch of samples.

Which is mathematical jargon for..."Trust me. It works, okay?"

So a sample of 1,600 people gives you a margin of error of 2.5 percent, which is pretty darn good for a poll. (See Margin of Error for more details on that term, and on polls in general.) Now, remember that the size of the entire population doesn't matter here. You could have a nation of 250,000 people or 250 million and that won't affect how big your sample needs to be to come within your desired margin of error. The Math Gods just don't care.

Of course, sometimes you'll see polls with anywhere from 600 to 1,800 people, all promising the same margin of error. That's because often pollsters want to break down their poll results by the gender, age, race or income of the people in the sample. To do that, the pollster needs to have enough women, for example, in the overall sample to ensure a reasonable margin or error among just the women. And the same goes for young adults, retirees, rich people, poor people, etc. That means that in order to have a poll with a margin of error of five percent among many different subgroups, a survey will need to include many more than the minimum 400 people in the overall sample.